The Annals of Probability

Stabilizability and percolation in the infinite volume sandpile model

Anne Fey, Ronald Meester, and Frank Redig

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Abstract

We study the sandpile model in infinite volume on ℤd. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure μ, are μ-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In d=1 and μ a product measure with density ρ=1 (the known critical value for stabilizability in d=1) with a positive density of empty sites, we prove that μ is not stabilizable.

Furthermore, we study, for values of ρ such that μ is stabilizable, percolation of toppled sites. We find that for ρ>0 small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.

Article information

Source
Ann. Probab. Volume 37, Number 2 (2009), 654-675.

Dates
First available in Project Euclid: 30 April 2009

Permanent link to this document
http://projecteuclid.org/euclid.aop/1241099924

Digital Object Identifier
doi:10.1214/08-AOP415

Mathematical Reviews number (MathSciNet)
MR2510019

Zentralblatt MATH identifier
1165.60033

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J25: Continuous-time Markov processes on general state spaces 60G99: None of the above, but in this section

Keywords
Abelian sandpile stabilizability percolation phase transition toppling procedure

Citation

Fey, Anne; Meester, Ronald; Redig, Frank. Stabilizability and percolation in the infinite volume sandpile model. Ann. Probab. 37 (2009), no. 2, 654--675. doi:10.1214/08-AOP415. http://projecteuclid.org/euclid.aop/1241099924.


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